I have that $Y_{ij} \sim N(\mu_i,\sigma)$ with $i=1,\dots,p$, $j=1, \dots,n$ and $\mu_i\mid\sigma^2 \sim N(\alpha_i, \frac{\sigma^2}{b})$, $\sigma^2 \sim GI(\gamma, \beta)$
I know that: $$P(\mu_i, \sigma^2\mid\textbf{y}) \propto P(\textbf{y}\mid \mu_i, \sigma^2)P(\mu_i, \sigma^2)$$
and $$P(\mu_i\mid\textbf{y}, \sigma^2, \mu_{(-i)}) \propto (\sigma^2)^{-\frac{np}{2}} \exp \left[-\frac{1}{2\sigma^2} \left( \sum_{i=1}^n (y_{ij}-\mu_i)^2+b(\mu_i-\alpha_i)^2 \right)\right]$$
So: $\mu_i\mid\textbf{y}, \sigma^2 \sim N(\mu^*, {\sigma^2}^*)$
$$\mu_i^*= \frac{(\frac{\sigma^2}{b})^{-2}\alpha +np\sigma^{-2}\bar{y}}{(\frac{\sigma^2}{b})^2 +np\sigma^{-2}} \text{ and } {\sigma^{-2}}^* = (\frac{\sigma^2}{b})^{-2}+np\sigma^{-2}$$
But i don't think the specification of $\mu_i^*$ and ${\sigma^2}^*$ is right.